International Journal of Number Theory

For a prime p, let ζ_p denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field k with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over k(ζ_p). In this paper, for p = 5 and 7, we study the ramification of such Kummer extensions using explicit Kummer generators directly computed by Verdure in 2006.

In [K. Girstmair, A criterion for the equality of Dedekind sums mod ℤ, Int. J. Number Theory10 (2014) 565–568], it was shown that the necessary condition b ∣ (a₁a₂ - 1) × (a₁ - a₂) for equality of two Dedekind sums s(a₁, b) and s(a₂, b) given in [S. Jabuka, S. Robins and X. Wang, When are two Dedekind sums equal? Int. J. Number Theory7 (2011) 2197–2202] is equivalent to 12s(a₁, b) - 12s(a₂, b) ∈ ℤ. In this paper, we give a new proof of this result and then find two additional necessary and sufficient conditions for 12s(a₁, b) - 12s(a₂, b) ∈ 2ℤ, 4ℤ. These give new necessary conditions on equality of Dedekind sums.

Ever since Erdős introduced the concept of a covering system in 1950, many questions have arisen regarding the existence of certain types of covering systems. Two of the most famous questions regarding covering systems are the odd covering problem and the minimum modulus problem. In the current paper, we ask two questions that are related to these famous questions and provide results for each.

Silverman and Stange defined the notion of an aliquot cycle of length L for a fixed elliptic curve E/ℚ, and conjectured an order of magnitude for the function that counts such aliquot cycles. We show that the conjectured upper bound holds for the number of aliquot cycles on average over the family of all elliptic curves with short bounds on the size of the parameters in the family.

Let pod_-3(n) denote the number of partition triples of n where the odd parts in each partition are distinct. We find many arithmetic properties of pod_-3(n) involving the following infinite family of congruences: for any integers α ≥ 1 and n ≥ 0,

A constructive generalization of Fleck-type Möbius function and inversion via convolution polynomials is presented here. Also provided are certain related propositions and some particular examples including a few classical results as consequences.

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves x^2m + ax^m + ay^m + y^2m = b whenever the ranks of some companion hyperelliptic Jacobians are at most one. As an application, we explicitly describe X_d(ℚ) for certain d ≥ 3, where X_d : T_d(x) + T_d(y) = 1 and T_d is the monic Chebychev polynomial of degree d. Moreover, we show how this later problem relates to orbit intersection problems in dynamics. Finally, we construct a new family of genus 3 curves which break the Hasse principle, assuming the parity conjecture, by specifying our results to quadratic twists of x⁴ - 4x² - 4y² + y⁴ = -6.

For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.

We obtain simple relations between the Hankel determinants of the formal power series ∏_k≥0(1 + Jx^3k) where , and prove that the sequence of Hankel determinants is an aperiodic automatic sequence taking value in {±1, ±J, ±J²}. This research is essentially inspired by the works about Hankel determinants of Thue–Morse-like sequences by Allouche, Peyrière, Wen and Wen (1998), Bacher (2006) and the first author (2013). It is worth mentioning that we do not have a unified result, even though Bacher's result and ours are very similar.

For a real number t, let s_t be the multiplicative arithmetic function defined by for all primes p and positive integers α. We show that the range of a function s_-r is dense in the interval (0, 1] whenever r ∈ (0, 1]. We then find a constant η_A ≈ 1.9011618 and show that if r > 1, then the range of the function s_-r is a dense subset of the interval if and only if r ≤ η_A. We end with an open problem.

Recall that a Dirichlet character is called imprimitive if it is induced from a character of smaller level, and otherwise it is called primitive. In this paper, we introduce a modification of "inducing to higher level" which causes imprimitive characters to behave primitively, in the sense that the properties of the associated Gauss sum and the functional equation of the attached L-function take on a form usually associated to a primitive character.

Assuming Dickson's conjecture, we give explicit formulas for Bhargava's factorials associated to the set ℙ₂ of twin primes. We also characterize the polynomials of degree n which take integral values on ℙ₂ by means of their values on {1, …, 2n + 1}. Doing this, we give a negative answer to a conjecture of Mingarelli. Similar results are also obtained when replacing ℤ by 𝔽_q[T].

It is shown that the quartic Fermat equation x⁴ + y⁴ = 1 has nontrivial integral solutions in the Hilbert class field Σ of any quadratic field whose discriminant satisfies -d ≡ 1 (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in , for p (> 7) a prime congruent to 7 (mod 8), but does have a nontrivial solution in the odd degree extension Σ of K. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in detail and the results are applied to prove several properties of the Weber singular moduli introduced by Yui and Zagier.